Trading Networks
Lada Adamic, Celso Brunetti, Jeffrey H. Harris, and Andrei Kirilenko*
Abstract
In this paper we analyze the time series of 12,000+ networks of traders in the emini S&P 500 stock index futures contract and empirically link network variables
with financial variables more commonly used to describe market conditions. We
show that network variables lead trading volume, intertrade duration, effective
spreads, trade imbalances and other market liquidity measures. Network variables
reflect information, information asymmetry and market liquidity and significantly
presage future market conditions prior to volume or liquidity measures. We also
find two-way Granger-causality between network variables and both returns and
volatility, highlighting strong feedback between market conditions and trading
behavior.
JEL Classifications: D85 Network Formation and Analysis, G12 Trading Volume,
G17 Financial Forecasting and Simulation, L14 Networks
*Adamic is with the University of Michigan, Brunetti is with the Board of Governors at the Federal
Reserve Bank, Harris is with Syracuse University, and Kirilenko is with the Commodity Futures
Trading Commission. We are grateful to Paul Tsyhura for invaluable assistance with the retrieval,
organization, and processing of transaction-level data. We thank Kirsten Anderson, Ana Babus, Matt
Elliott, Pat Fishe, Ben Golub, Matt Harding, Matt Jackson, Pete Kyle, Shawn Mankad, Antonio
Mele, Anna Obizhaeva, Han Ozsoylev, Kester Tong, Kumar Venkataraman and presentation
participants at the American University, Cambridge University, the Chicago Mercantile Exchange,
the Commodities Futures Trading Commission, 2009 Complexity Conference at Northwestern
University, 2009 Econometric Society Summer Meetings in Barcelona, the Federal Reserve Board of
Governors, NASDAQ, 2009 NBER-NSF Time Series Conference, Oxford, the Securities and
Exchange Commission, Southern Methodist University, Stanford University, Syracuse University,
the University of Maryland, the University of Michigan, Villanova University, and 2009 Workshop
on Information in Networks at New York University for very helpful comments and suggestions.
Some earlier drafts of the paper were circulated under the title ”On the Informational Properties of
Trading Networks.“ The views expressed in this paper are our own and do not constitute an official
position of the Commodity Futures Trading Commission, its Commissioners or staff.
Financial markets bring buyers and sellers together, aggregating information
for price discovery and providing liquidity for uninformed, liquidity-seeking order
flow. Indeed, numerous models show that trading outcomes – prices, volume,
volatility, and effective spreads – depend on the complex mix of information flows
and liquidity demands of individual buyers and sellers.1 In this paper we use
established network analysis tools to characterize information and liquidity flows in
financial markets. We find strong and significant correlations between network
statistics and financial variables, evidence that quantitative network parameters
capture information, information asymmetry and liquidity characteristics in the emini S&P500 futures market.
Moreover, we find that the time series of network statistics significantly
presage (Granger-cause) intertrade duration, trading volume, effective spreads,
trade imbalances and signed volume imbalances. Since duration and volume have
been shown to proxy for information and liquidity in financial markets,2 these
findings suggest that network statistics serve as primitive measures of information
and liquidity that emerge prior to the advent of trading volume and other market
liquidity changes. In this regard, network statistics contain useful information
beyond that contained in past volume and other more standard market descriptors.
We also find that network variables capture complex feedback mechanisms
that likely underlie high frequency trading strategies. Our network variables
See, for instance, Copeland and Galai (1983), Glosten and Milgrom (1985), Kyle (1985), Admati and Pfleiderer
(1988), Foster and Viswanathan (1990, 1993, 1994). The theoretical literature on limit order markets includes
Parlour (1998), Foucault (1999), Biais, Martimort and Rochet (2000), Parlour and Seppi (2003), Foucault, Kadan
and Kandel (2005), Back and Baruch (2007), Goettler, Parlour, and Rajan (2005, 2009), Large (2009), Rosu
(2009), and Biais and Weill (2009), among others.
2 See Epps and Epps (1976) and Engle and Russell (1998).
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exhibit bi-directional Granger-causality with returns and volatility measures,
highlighting the fact that trading strategies evolve dynamically in response to
changing market conditions. Importantly, we replicate the contemporaneous
correlations between financial and network variables in a simulated agent-based
trading model with randomly generated orders, but find no Granger-causality in
simulated networks. Thus, the fact that network variables lead volume, duration
and liquidity metrics does not stem from some statistical artifact of network
construction, but rather from real market interactions.
Our sample consists of more than 7.22 million transactions conducted during
August 2009 in the nearby e-mini S&P 500 futures contract. Utilizing proprietary
data from the U.S. Commodity Futures Trading Commission (CFTC) that identifies
the trading account on both sides of every trade, we reconstruct the links between
buyers and sellers over short intervals of time, creating a time series of more than
12,000 networks each comprised of 600 sequential trades. Importantly, these
financial networks reflect the microstructure of the U.S. equity index futures
markets that serve as our laboratory.
Our results are robust to different equity index futures markets (e-mini Dow
Jones and Nasdaq 100 futures), different observation periods (May and August
2008), and different sampling frequencies (240 and 600 transactions). The
juxtaposition between live and simulated markets suggests that these Grangercausality results stem from strategic trading behavior and not from some statistical
artifact of network construction.
2
While graph theory dates back several centuries, more recent applications
offer novel insights across a diverse range of disciplines (see Newman (2010)).
Network analysis yields insight into molecular processes in cells, the stability of
ecosystems, and epidemic thresholds in human contact networks. Likewise, network
analysis is fruitful in modeling large scale information systems (such as the
Internet) and devising algorithms for efficient information retrieval (such as
Google). Applications in economics and finance range from analyses of interactions
among firms to correlations between various asset price time series.3
In the network literature, most applications to human social networks model
the interaction between entities that drive overall network dynamics. Since the
clearinghouse is the counterparty to all futures trades, our networks represent
traders matched on straightforward price and time priority rules. In our work,
traders interact only through the trade matching process in the limit order book and
do not intend to necessarily trade with each other. We show that network metrics
can capture many salient features of financial markets, from localized patterns of
trade to market-wide characteristics.
In finance, a growing literature addresses banking networks. Leitner (2005)
models bank networks where linkages both spread contagion and induce private
sector bailouts. Similarly, Babus (2009) models the contagion risk that a bank
Other financial network papers include Caldarelli and Catanzaro (2004), who analyze networks of corporate
boards, Jackson (2007, 2009), Allen and Babus (2008), Colla and Mele (2009), and Oszoylev and Walden (2009)
who summarize network applications in economics and finance and Onnela, Kaski and Kertész (2004), Boginski,
Butenko and Pardalos (2005) and Castiglionesi and Navarro (2010), Fenn, Porter, Williams, McDonald, Johnson
and Jones (2010) who model financial networks more generally.
3
3
failure propagates through networks. Allen, Babus and Carletti (2010) also model
bank portfolio decisions as a network formation game.4
This is the first paper to empirically examine the links between network
statistics and standard financial variables. We consider trading network
parameters as primitive descriptors of the order execution process – not outcomes
resulting from the preferences of individual traders, but rather from the
simultaneous and aggregate liquidity demands and information-based trade that
comprise financial markets.
Our results suggest that network statistics and
standard financial descriptors are strongly linked economically, with various
network statistics serving primary roles in describing information and liquidity in
financial markets. In this regard, network statistics might be useful for
constructing profitable trading strategies.
The paper proceeds as follows. In Section I, we describe our unique high
frequency data and our network and financial variables. In Section II, we show
examples of different trading networks and use these examples to illustrate how our
network statistics relate to financial variables. In Section III, we present
contemporaneous correlations and lead-lag (Granger-causality) tests among
network and financial variables. Section IV presents an agent-based simulation
model of trading networks in order to compare our empirical market results to
results in a controlled setting. We conclude with Section V.
Other recent papers on trading networks include Blume, Easley, Kleinberg and Tardos (2009) and Cohen-Cole,
Kirilenko and Patacchini (2011).
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1. High Frequency Data, Network Metrics, and Financial Variables
We use audit trail, transaction-level data for over 7.2 million regular
transactions (executed between 9:30 a.m. ET and 4:00 p.m. ET) in the September
2009 e-mini S&P 500 futures contract during August 2009. The e-mini S&P 500
futures contract is a highly liquid, fully electronic, cash-settled contract traded on
the CME GLOBEX trading platform.5 Each transaction includes the date, time (up
to the second), unique transaction identifier (which enables us to order transactions
sequentially within each second), executing and opposite trading accounts,
executing and opposite brokers, buy/sell flag, price, and quantity.6 31,585 unique
accounts trade during our August 2009 sample.
Since our goal is to explore links between financial variables and network
statistics, we first determine a sampling frequency that ensures our financial
variables are not contaminated by market microstructure noise. We apply both the
Andersen, Bollerslev, Diebold and Labys (2000) volatility signature plot and Bandi
and Russell (2006) technique and find that the smallest acceptable sampling
window must contain at least 50 transactions. However, networks constructed over
such a small number of transactions can be too sparsely connected to adequately
capture the dynamic nature of the informational and liquidity forces that drive
markets. Consequently, we use 600 transactions as a conservative sampling window
to construct meaningful trading networks and to compute network statistics.
Although pro rata allocations and lead market makers exist on the CME, GLOBEX trades in the e-mini S&P
contract are matched with price-time priority.
6 We applying standard filters designed to look for recording errors and outliers in the price and quantity series
(see Hansen and Lunde (2006)) and find no irregularities in the data.
5
5
1.1 Networks and Network Metrics
Over the entire month of trading, we construct 12,032 sequential (nonoverlapping) trading networks, each comprised of 600 consecutive trades.7
Following network terminology, a network consists of nodes and edges. We define a
node as an individual trader (trading account) and an edge that connects a pair of
nodes as a trade between two traders. In our work the edges are directed, with buys
representing edges pointed toward a trader and sells representing edges pointed
away from a trader. Multiple transactions between traders are represented by a
single directed edge pointed toward the buyer and away from the seller, even if the
prices and quantities of each transaction differ. Additionally, from network
terminology the degree of a node is the number of edges connected to it, with
indegree (outdegree) representing the number of edges pointing toward (away from)
the node. For clarity, we use the terms buydegree and selldegree to represent
indegree and outdegree, respectively.
Quantitative analysis of networks employs a set of standard metrics. We
focus on network metrics that represent the properties of information and liquidity
in markets. Centrality, for instance, quantifies the importance of a specific trader in
a network. Although several centrality measures exist, we use degree centrality to
characterize how “central” a particular trader is in terms of its trading, utilizing the
buy/sell indicators in our data to construct buydegree and selldegree centrality–
Since the number of trades during the trading hours is seldom precisely divisible by 600, we complete the final
network for each day with the trades reported after the close. Our results are robust to excluding this final
network of each day.
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representing the number of bilateral purchases and sales, respectively, for each
trader in the network.8
We then compose aggregate centralization from individual trader centrality
measures that characterizes the inequality in degree among all traders in the
network. Specifically, we compute a centralization Gini (GB,S) for buyers (B) and
sellers (S) defined as:
GB,S =
∑𝑁
1 (2𝑟𝑖 − 𝑁 − 1)𝑘𝑖
𝑁∗𝐸
summing over all N traders, where ki is the ith trader's buydegree and selldegree, ri,
i=1,…,N, is each trader's rank order number9 and E is the number of unique
connections between traders present among the 600 trades in each network. By
construction, GB and GS are 0 if every trader has the same number of buy and sell
connections,
and
positive
with
increasing
degree
inequality.
Maximum
centralization occurs when one trader does all the buying or selling in the network.
Using the above formula, we compute network-wide centralization as the
difference between GB and GS. Intuitively, centralization can be interpreted as the
presence of a dominant buyer (close to 1) or a dominant seller (close to -1) within the
600-trade network. The absolute value of centralization (|centralization|) therefore
represents the presence of a dominant trader on either side of the market. In
economic terms, centralization represents a measure of informed trading or strong
net demand for liquidity in the market.
Note that buydegree and selldegree do not fully capture the role of a trader in the network. For instance, a
trader that does not execute a large number of trades but serves to connect otherwise disconnected traders may
be very central, but will have relatively low buydegree and selldegree measures.
9
The trader with the highest centrality ranks highest, the trader with the second highest centrality ranks
second, and so on. We rank buydegree and selldegree centrality separately.
8
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Table I presents summary statistics for the network variables across all
12,032 networks in our sample.10 Centralization is approximately symmetric and
ranges from -0.74 to +0.65 with a mean near zero and standard deviation of 0.17.
|Centralization| ranges from 0.0 to 0.74 with mean 0.14 and standard deviation of
0.10. The mass of this distribution is more than one standard deviation away from
zero, which can be interpreted as inequality in the number of buy vs. sell matches
per trader.
Assortativity in networks represents the tendency of `like to be connected
with like' for any node (Newman (2002)). For our purposes, we use each trader's
buydegree and selldegree to assess assortativity within our networks, examining
the propensity of highly connected buyers to trade with highly connected sellers, for
instance. In an assortative network, buyers/sellers with many edges (high
buydegree/selldegree nodes) are more likely to connect to similar buyers/sellers. In a
disassortative network, buyers/sellers with many edges tend to connect to
buyers/sellers with few edges. Intuitively, assortativity represents a measure of
asymmetric information or liquidity imbalance in a trading network. 11 For instance,
one large buyer matched with a number of small sellers exhibits high assortativity
while a large buyer matched with a single large seller exhibits low assortativity.
We measure assortativity by the Pearson correlations between buydegree and
selldegree for all edges present in the network. Using the notation above, over all
Each network variable is stationary and exhibit persistent autocorrelations. Jarque-Bera (1980) tests reject
the null of normality at standard significance levels.
11 Glosten (1987), Stoll (1989), George, Kaul and Nimalendran (1991) and Madhavan, Richardson and Roomans
(1997) each model asymmetric information components in quoted bid-ask spreads.
10
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edges Ei.j we calculate four pairwise correlations ρ(kibuy, kjbuy), ρ(kibuy, kjsell), ρ(kisell,
kjbuy), and ρ(kisell, kjsell) corresponding to the four conditional degree distributions
(buydegree ki,jbuy and selldegree ki,jsell) for each connected trader pair i and j.
Intuitively, the coefficient ρ(kisell, kjbuy) measures the correlation between the
number of unique buyers to whom a seller is selling to and the number of unique
sellers that those buyers are buying from. A negative correlation indicates that
when a seller matches with many buyers, those buyers are buying from few or no
other sellers.
From these four correlations, we construct an aggregate assortativity index
(AI) for each network
𝐴𝐼 =
1
𝑏𝑢𝑦 𝑏𝑢𝑦
𝑏𝑢𝑦
𝑏𝑢𝑦
(𝜌(𝑘𝑖 , 𝑘𝑗 ) − 𝜌(𝑘𝑖 , 𝑘𝑗𝑠𝑒𝑙𝑙 ) − 𝜌(𝑘𝑖𝑠𝑒𝑙𝑙 , 𝑘𝑗 ) + 𝜌(𝑘𝑖𝑠𝑒𝑙𝑙 , 𝑘𝑗𝑠𝑒𝑙𝑙 ))
4
computed over all edges, where the scaling factor, ¼, assures that the assortativity
index falls between -1 and 1. By construction, the assortativity index captures
patterns in networks that feature large degree dispersion. For example, the
assortativity index is high in a network that contains a trader (like a dominant
intermediary) who is both a large buyer and a large seller trading with many small
counterparties. Likewise, the assortativity index is also large in a network that has
a large buyer and a large seller connected through many small intermediaries. In
either example, trades occur between parties that differ in both connectivity and
liquidity provision/removal.
As Table I shows, the assortativity index in our sample ranges from -0.06 to
+0.34, with a mean of 0.04 and standard deviation of 0.04. This positive mean
9
stems in part from the skewed degree distributions. Most buyers have low
buydegree and, on average, buy from a seller with high selldegree. Similarly, most
sellers have low selldegree and, on average, sell to buyers with high buydegree. This
pattern is consistent with a market populated by a small number of highly
connected intermediaries who buy and sell from many liquidity-demanding (or
informed) traders.
Clustering is a measure of transitivity in the network, i.e. if i trades with j,
and j trades with k, clustering measures whether i also trades directly with k. We
quantify clustering using the global clustering coefficient (CC) (Newman (2002)):
𝐶𝐶 =
3 ∗ 𝑇𝑐𝑙𝑜𝑠𝑒𝑑
𝑇
where T represents the total number of “connected triples” of three traders (i, j and
k) and Tclosed represents the number of “closed triples” where i trades with j, j trades
with k, and i also trades directly with k.12
Economically, the clustering coefficient represents liquidity in the market.
Connected triples represent the presence of at least one short-term intermediary.
Intuitively, large clustering coefficients represent greater liquidity. In the extreme
case where a single trader provides liquidity, no closed triples exist and the
clustering coefficient is zero. At the other extreme, where all triples involve three
traders connected as a “closed triple”, the clustering coefficient is 3.
As shown in Table I, the clustering coefficient in our sample ranges from 0.0
to 0.23, with an average of 0.05 and standard deviation of 0.03. On average, there is
We treat the edges as undirected in the clustering coefficient since intermediation/liquidity involves both
buying and selling. Fagiolo (2007) discusses directed clustering coefficients.
12
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only a slight tendency for traders to cluster in this market. Indeed, the average
clustering coefficient we observe is nearly identical to the coefficient we would
expect by randomly connecting traders with 600 trades.
Dispersion of information or liquidity in the market may be measured using
connected components. A connected component is the set of all traders connected
with each other through bilateral trading. The largest strongly connected
component (LSCC) is the maximum number of traders that can be reached from any
other trader by following directed edges.
We compute the largest strongly connected component as:
𝐿𝑆𝐶𝐶 =
𝐿𝑆𝐶𝐶𝑀𝑎𝑥
𝑁
where LSCCmax is the raw count of traders in the LSCC and N is the total number
of traders in the market. This ratio ranges between 1/N (one trader connects all
other traders) and 1 (all traders are connected to all other traders).
A larger LSCC forms when many traders are both buying and selling, that is
when the supply of liquidity is dispersed among many traders. For example, a large
strongly connected component is likely to emerge as a result of a large number of
limit orders rather than one large market order. Since the LSCC is scaled by the
total number of traders in the market, the arrival of concentrated net demand for
liquidity will manifest itself in a smaller LSCC. As Table I indicates, the portion of
the network occupied by the largest strongly connected component varies
significantly, ranging from 0.00 to 0.52, with a mean of 0.09 and standard deviation
of 0.06.
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1.2 Financial Variables
For each of the 12,032 sampling periods, we also compute market returns,
volatility, intertrade duration, trading volume, and more standard liquidity
measures including effective bid-ask spreads, and signed volume, buy/sell trade
imbalances. These variables are typically utilized in economic studies to describe
financial market conditions. Lastly, given the fact that our clustering and
centralization measures may be related to market concentration, we also construct
a Herfindahl measure of buy/sell trade concentration during each interval.
Descriptive statistics for these financial variables are shown in Table II.13
Market returns contain valuable information about the true underlying price
formation process, but may also contain market microstructure noise, measurement
errors, and seasonal patterns (Engle (2000)). We compute open-to-close market
returns as the log difference between the last and the first transaction price for each
network period. We remove the predictable intraday seasonal component from raw
returns by regressing returns on a constant and a sequence of dummy variables for
each half-hour during the trading day and then use the unexplained term as our
measure of market returns.14 As shown in Table II, returns range from -0.15 to
+0.20 and are centered on zero with standard deviation of 0.04. Returns exhibit
slightly negative skewness.
As with our network variables, all financial variables in our sample are stationary and (other than returns)
highly persistent, with significant autocorrelation coefficients at 1, 5, and 10 lags.
14 We apply the same technique and use a Fourier flexible form to remove seasonality from all variables and
examine alternative close-to-close returns with similar results.
13
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We use three measures to estimate volatility during each sampling interval
period: absolute returns, squared returns, and the log difference between the
maximum and minimum prices (price range). For the results reported below, we use
price range as an estimate of volatility.15 Importantly, however, our main results
are not affected by the choice of volatility estimator. As shown in Table II, across
12,032 sampling intervals, volatility averages about 0.06 percent, corresponding to
an annual volatility of 23.56 percent. Volatility ranges from 0.02 to 0.29 percent,
with a standard deviation of just 0.02 percent.
Trading volume contains valuable information about the underlying price
formation process, because volume together with observed transaction prices may
be driven by a common latent factor.16 We compute trading volume as the number of
contracts both bought and sold during each network interval. Volume ranges from
1095 to 7706 with an average of 2629 and standard deviation of 636 contracts.
Intertrade duration can be interpreted as a proxy for the arrival of new
information or liquidity to the market (Engle and Russell (1998) and Engle (2000)).
We compute three measures of duration: total period duration, volume-weighted
period duration, and the average of the 599 intertrade durations and report results
for total period duration – the time elapsed between the start and end of the
network interval. Our main results are not affected by the choice of duration
Range-based volatility estimators are more efficient than return-based volatility estimators (see, e.g.,
Parkinson (1980), Garman and Klass (1980), Beckers (1983), and Brunetti and Lildtholdt (2006)). Christensen
and Podolskij (2007) use the price range to compute realized volatility in high frequency data.
16 The vast literature on the properties of trading volume includes Clark (1973), Epps and Epps (1976), Tauchen
and Pitts (1983), Admati and Pfleiderer (1988), Easley and O'Hara (1992), and Andersen (1996), among others.
15
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estimator. As shown in Table II, the intertrade duration ranges from zero to 344
seconds. In our sample, 600 transactions occur every 40.8 seconds, on average.
Liquidity reflects the ease with which a security can be bought or sold
without a significant price effect. Unlike trading volume and intertrade duration,
liquidity is not directly observable and has multiple dimensions that are difficult to
capture in a single measure. Indeed, as discussed above, our network variables
capture some of these dimensions. We also measure more traditional liquidity
metrics, like effective spreads, signed volume and trade imbalances.17 The effective
spread captures the implied cost of trading and is equal to twice the square root of
the first order autocovariance of returns over each interval (Stoll (1978)). Signed
volume (buy minus sell volume) and trade imbalances (number of buys minus sells)
capture the propensity for buy and sell orders to match up during the interval,
another dimension of liquidity.
The e-mini S&P 500 contract is extremely liquid. Table II shows that the
average effective spread is less than 0.8 cents, with a maximum effective spread of
just over two cents during our sample period. The raw Amihud illiquidity measure
(unreported) also has mean, median, and standard deviation very close to zero and
by this measure, it takes about 21 contracts (or over $1 million) to move prices by
one tick (0.25 index points). While mean and median signed volume and trade
imbalances are very small, the market can be imbalanced, with maximum signed
volume exceeding 5000 contracts and maximum buy imbalance at 13 trades. These
We also compute Amihud illiquidity measures using both contract volume and dollar trading volume with
nearly identical results.
17
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extreme cases suggest that market liquidity can be stressed during the short
intervals we study, giving us reason to explore whether liquidity metrics might also
presage returns, volatility and volume below.
Table II also displays a Herfindahl measure computed over each trading
interval to compare with the clustering and centralization metrics we apply from
network statistics. As shown, the Herfindahl trades statistic is less variable than
most other variables, with a minimum of 0.0002, maximum of 0.0013 and standard
deviation of just 0.0001.18
2. Illustrative Examples
Prior to conducting the time-series statistical analysis for the full sample of
the network and financial variables, we present statistics from representative
trading networks to illustrate the key concepts and the intuition behind our
approach. Figure 1 presents three actual trading networks from our data,
accompanied by network statistics and financial variables. The three trading
networks are chosen to display relatively high values of centralization (left column),
assortativity (middle column), and both clustering and the largest strongly
connected component (right column).
The left column of Figure 1 presents a network with one dominant seller
(perhaps an informed trader or trader with strong liquidity demand) matched with
many small buyers. As shown in Panel A, this trading network has a large negative
centralization coefficient, intermediate levels of assortativity and LSCC, and a
18
In unreported results we also construct a Herfindahl volume metric with nearly identical results.
15
small clustering coefficient. As shown in Panel B, this network is also associated
with a large negative market return, high volatility, intermediate trading volume,
and low duration. Liquidity, volatility, duration and return statistics suggest that
strong liquidity demand drains a significant portion of market liquidity and made a
considerable price impact. Indeed, the signed volume and trade imbalance are
accompanied by a large negative return, volatility, and short intertrade duration
suggesting a large market order which quickly “walks the book.”
The center column of Figure 1 presents a trading network with a relatively
large buyer trading with several moderately-sized sellers through a number of
intermediaries. As shown in Panel A, this network is characterized by intermediate
centralization and clustering levels, suggesting that the net demand for liquidity is
relatively balanced, with slightly negative signed volume but positive trade
imbalance. Moreover, the high assortativity index means that large traders are
mostly matched with many small traders (rather than with each other); and the
small LSCC suggests that liquidity is provided by just a few intermediaries. As
shown in Panel B, this network is associated with good liquidity and relatively low
effective spreads. Furthermore, this trading network is characterized by high
intertrade duration and relatively low volume, so these 600 trades are relatively
small and executed quickly. This network is associated with intermediate positive
market returns and intermediate volatility levels. Overall, these market conditions
represent a handful of intermediaries supplying liquidity to smaller traders
resulting in low volatility, low effective spreads and slightly positive returns.
16
The right column of Figure 1 presents a trading network with greater
dispersion among buyers and sellers of various sizes, reflected in the zero
centralization metric (no dominant buyer or seller exists) and low assortativity
(“like traders trade with like”). High levels of clustering and LSCC, suggest that the
net supply of liquidity is dispersed among many. Indeed, this trading network
exhibits zero returns accompanied by good liquidity–low signed volume imbalance,
zero trade imbalance and relatively large trading volume.
These three examples illustrate our conjecture that network statistics and
standard financial variables capture different dimensions of market conditions.
Indeed, since information and liquidity appear to drive both network and financial
variables, the two sets of variables are likely to be statistically interrelated as well.
We formally examine the statistical relation between network metrics and
traditional financial variables below.
3. The Statistical Relation between Network and Financial Variables
We first examine contemporaneous correlations among network and financial
variables and then test for lead-lag relations between and among these variables.
Our ultimate goal is to discover whether network analysis tools can serve to
improve our understanding of financial market conditions beyond what more typical
financial descriptors provide.
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3.1 Correlations
Panel A in Table III reports contemporaneous correlations among network
variables. Although centralization is not correlated with other network variables,
the other network variables are often strongly contemporaneously correlated with
each other. |Centralization| is negatively correlated with both the clustering
coefficient and the LSCC. Intuitively, this means a dominant trader increases net
demand for market liquidity and counterparties to that dominant trader are less
likely to trade with each other. Conversely, the high positive correlation between
clustering and LSCC suggests that dispersed net liquidity supply among many
traders is often accompanied by strong liquidity demand. The assortativity index is
relatively uncorrelated with centralization measures and clustering, but is strongly
negatively correlated with the LSCC. Intuitively, both higher assortativity and
lower LSCC reflect greater liquidity imbalance.
Panel B in Table III examines this economic intuition more closely with
contemporaneous correlations between financial and network variables. Consistent
with our illustrative examples above, centralization is strongly correlated with
returns, signed volume and trade imbalances. At the same time, while
|centralization| is positively correlated with volume and volatility, it is negatively
correlated with duration, effective spreads and trade concentration. Intuitively, a
large order (with high |centralization|) has a significant chance to walk the limit
order book, resulting in higher volatility and volume with lower intertrade duration.
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Higher
asymmetric
information
or
greater
liquidity
imbalance,
as
represented by the assortativity index, is negatively correlated with volume and
volatility, but largely unrelated to more traditional market metrics. The former
suggests that higher asymmetric information or liquidity imbalance is accompanied
by a slight reduction in both volatility and trading volume. The latter, however,
suggests that asymmetric information, as represented by assortativity, is not
related to signed volume or trade imbalances.
Clustering is positively correlated with volume and negatively correlated
with effective spreads and volatility. Intuitively, since clustering captures shortterm market making activities, greater intermediation improves liquidity, smooths
volatility and enhances trading volume. Clustering is also highly negatively
correlated with trade concentration (Herfindahl trades). The LSCC, the dispersion
of net liquidity supply, is positively correlated with both volume and volatility. The
LSCC, however, is not significantly correlated with other more traditional financial
market variables.
3.2 Lead-Lag Relations
Correlations offer evidence that some network and financial variables are
contemporaneously related and suggests that network variables may be redundant
descriptors of market conditions. In this section, we examine the lead-lag relations
among network and financial variables to gauge the incremental value in
19
calculating non-standard network metrics. As noted above, we conjecture that
network metrics serve as primitive measures of information and liquidity.
To explore whether network variables presage or follow standard financial
variables we apply standard Granger causality tests in vector autoregressive (VAR)
models.19 Table IV provides the p-values of Granger-non-causality tests among the
five network variables. Centralization neither Granger-causes nor is Grangercaused by other network variables (p-values equal to 0.65 and 0.37, respectively). As
a system, however, the remaining network variables are both jointly Grangercaused by (and jointly Granger-cause) each other (p-values between 0.00 and 0.05),
indicating strong feedback effects. Indeed, most pair-wise tests are also significant.
Table V presents p-values for the Granger-non-causality tests between
network variables and each financial variable both jointly and independently. In
Panel A, we find that returns and volatility are jointly both Granger-caused by and
Granger-cause network variables (p-values of 0.00 and 0.01 for returns and 0.00
and 0.00 for volatility, respectively). Centralization drives the result for returns,
with strong bi-directional Granger-causality. For volatility, we find strong bidirectional Granger-causality (representing feedback effects) between volatility and
each of our four network variables independently.
Panel B in Table V reports test results for intertrade duration and volume.
We find strong evidence of one-way causality–duration and volume are both
Since the variables exhibit heteroskedasticity and serial correlation, we estimate VAR models using
generalized method of moments (GMM) and Newey-West robust standard errors. Standard tests (available upon
request) show that the VAR model must include all five network variables. The Akaike and Schwartz
Information Criteria indicate an optimal lag-length of twelve.
19
20
Granger-caused by each network variable, both individually and jointly (joint pvalue = 0.00). While duration leads the LSCC and volume leads assortativity
independently, neither duration nor volume lead network variables jointly (p-values
= 0.20 and 0.16, respectively). These results indicate that network statistics presage
both duration and trading volume. To the extent that duration and volume proxy for
information arrival, network statistics reflect forthcoming changes in market
conditions.
Lastly, Panels C and D show that network variables jointly Granger-cause
signed volume, trade concentration, effective spreads and trade imbalances. These
strong results suggest that the demand and supply of liquidity are reflected in
network variables prior to emerging in more traditional liquidity measures. While
individual pair-wise tests vary significantly, we find no feedback effects here. Taken
as a whole, these network statistics serve to presage future, short-term changes in
both liquidity and market concentration.
For comparison, Table VI displays bi-variate VAR Granger-causality tests
between our liquidity and the traditional market statistics of returns, volatility,
volume and duration. Contrary to results obtained with network statistics, we find
bi-directional causality among many of these variables, suggesting that they are
jointly determined in the market during the same trading interval.
Effective
spreads, in particular, exhibit bi-directional relations with volatility, volume and
duration. Trade imbalances (Panel B) are unrelated to these other market statistics.
Interestingly, we find no lead-lag in either direction between these
21
traditional liquidity measures and returns, in stark contrast to the strong bidirectional effects between network statistics and returns. Centralization, in
particular, is strongly related to returns (Table V) while market concentration
(Table VI) is not significantly related to returns at all. These results confirm that
our network metrics capture an element of market conditions that is simply not
found in traditional market descriptors.
3.3 Robustness
The links between network and financial variables are remarkably robust
with respect to different markets, observation periods, sampling frequencies and
levels of aggregation (defining nodes as either individual accounts or executing
brokers). We replicate correlations and lead-lag results for both the e-mini Nasdaq
100 and e-mini Dow Jones stock index futures contracts. We also find qualitatively
similar results for these three equity futures contracts during the individual months
of May 2008, August 2008 and August 2009. The results also hold for networks
constructed with sampling windows of 240, 600 and 1,200 transactions during each
month, suggesting that our results are not sample-specific, market-specific, nor
specific to the trade construction of the network.
4. An Agent-based Simulation Model Benchmark
The bi-directional Granger-causality that we find between network and
financial variables suggests strong feedback effects in short intervals. In order to
explore the source of this feedback, we construct an agent-based simulation model
22
of a limit order market that is devoid of a feedback mechanism. The simulated
model allows for heterogeneous beliefs about the price process, but imparts no
intentionality or memory upon the traders. Consequently, we might expect to find
significant correlations among network and financial variables, but not necessarily
evidence of Granger-causality.
In the simulation we endow a fixed number of traders with orders which
arrive with a Poisson distributed inter-arrival time and are assigned at random.
The quantity for each order is drawn from a lognormal distribution and with an
equal probability of being a buy or a sell order. For a sell order the price is set a
small fixed number above the last transaction price plus a log-normally distributed,
mean zero random variable. For a buy order, the price is set a small fixed number
below the last transaction price plus a log-normally distributed, mean zero random
variable. The (log-normally distributed, zero mean) random variable added to the
order price is consistent with an equilibrium price function under the assumption of
heterogeneous beliefs about the true price process (see Scheinkman and Xiong
(2003)).
As our simulation starts, orders begin to populate the limit order book. Each
incoming order is compared to previously placed orders by price and time priority. If
a match between a buy and sell order is made, a transaction takes place. If an order
is only partially filled, we attempt to match the remaining quantity against other
orders on the book, and if no match is found, the order remains in the limit order
23
book. In order to avoid stale limit orders, each order expires and is withdrawn from
the market after 100 subsequent transactions are executed.
We simulate orders until we attain 7.2 million transactions, segment the data
into 12,000 networks of 600 consecutive transactions, and compute network and
financial variables for each sampling interval using the same methods applied to
the empirically observed data. We then analyze both correlations and lead-lag
relations among network and financial variables to compare with actual market
results.
Generally, we find that the simulated executions generate contemporaneous
correlations similar to, but less significant than, those obtained from the real
market data. The correlations between network variables and between network and
financial variables are displayed in Table VII. As with the market data, we find
strong correlations between returns and centralization. The simulation also sheds
light on the possible sources of this high correlation. We find the mechanics of the
impatient order submission strategy raises the correlation: when a large buy order
at a high price is matched against several existing sell orders network
centralization is high. Matching a large number of sell orders to a single buy order
is more likely to reflect the large buy order walking up the book, so
contemporaneous returns are also high.
Generally, correlations between volatility and |centralization|, and between
the Hefindhal index and |centralization| are also relatively high in the simulated
networks. These results suggest that the order matching process can generate
24
contemporaneous correlations between financial and network variables in the
absence of any economic motivation (other than the prescribed urgency to trade
built in to simulate perishable information).
However, although our simulations prescribe order execution rules and order
cancellation times, these simulations do not allow for dynamic trading strategies
that adapt to changing market conditions. As expected, Granger-causality tests
among simulated network and financial variables lack the dynamic structure found
in live market data.20 Indeed, Granger-causality tests among network and financial
variables are generally insignificant and yield few feedback effects, indicating a
very poor fit. This suggests that the Granger-causality results that we find in the
actual market data arise as a result of the strategic behavior of traders and are not
simply artifacts of the order matching process.
This evidence further suggests that network statistics uniquely characterize
information and liquidity dynamics in financial markets. While simulations
generate similar correlations between network and traditional market variables,
they cannot replicate the lead-lag relations between these variables. Further, the
strong Granger-causality running from network to traditional market variables in
real markets is not an artifact of the order matching process. Rather, these
simulation results suggest that the real market lead-lag relations reflect latent
order submission strategies in financial markets, strategies that manifest
Reflecting the lack of dynamics in the simulated data, the AIC (SIC) selects an optimal lag-length of order one
(zero) in the VAR specification. We use lag-length of order one.
20
25
themselves in the composition of the network prior to the advent of volume or
liquidity changes.
5. Conclusions
We use audit trail data that identifies trading accounts on both sides of every
trade to construct the links between buyers and sellers in the e-mini S&P 500 stock
index futures contract, the price discovery venue for the U.S. equity market. We
construct a time series of more than 12,000 trading networks each comprised of 600
sequential trades and use established network analysis tools, to empirically link
network variables with financial variables that describe market conditions. We find
that network statistics are contemporaneously correlated with these financial
variables, suggesting that the underlying economics that drive financial variables
also
drive
network
parameters.
Importantly,
network
variables
capture
information, information asymmetry and liquidity characteristics in the e-mini
S&P500 futures market that we study.
While network and financial market variables are contemporaneously
correlated, network variables are primitive determinants of market conditions,
emerging prior to many more common measures of information and liquidity. For
example, we find that network statistics strongly Granger-cause trading volume,
intertrade duration, effective spreads, trade imbalances and signed volume
imbalances. Moreover, we document bi-directional Granger-causality between
network variables and both returns and volatility, suggesting that network
26
variables also capture complex feedback mechanisms that underlie trading
strategies in financial markets..
We conjecture that market-wide patterns of order execution are informative
about the price formation process in order driven markets and these dynamics are
manifest in both financial and network variables. Given the fact that network
variables presage volume, duration and other market liquidity measures, we
conclude that trading network analysis offers a fruitful mechanism for assessing the
underlying strategies that drive price formation and liquidity supply in financial
markets.
Given the strong contemporaneous link between financial and network
variables, we believe that network analysis tools might also offer new insights into
the behavior of financial markets. The data we employ here differs from the typical
social network setting where connections are made intentionally. The trade
matching algorithm in these futures markets simply pairs up buyers and sellers
with price-time priority. Whether our results hold in settings where orders are
executed via preferencing arrangements, across competitive trade execution venues
(like the U.S. equity markets), in floor-based exchanges, or in markets where
internalization or payments/rebates for order flow are possible remain questions for
further research.
27
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32
Panel A: Network Metrics
0.044
0.000
0.044
0.000
0.297
0.011
0.009
0.170
0.005
0.333
Panel B: Traditional Financial Variables
Returns
-0.100
0.051
0.000
Volatility
0.100
0.051
0.051
Volume
2325
2304
3832
Duration
10
19
14
Effective Spread
0.007
0.005
0.009
Signed Volume
-2151
-557
-170
Trade Imbalance
-4.0
2.0
0.0
Herfindahl Trades
7.50
7.94
9.56
Figure 1. Trading Network Statistics This figure displays three examples of trading networks comprised of 600 sequential
trades in the nearby e-mini S&P 500 futures contract on the CME Globex platform during August 2009. Centralization,
|Centralization|, assortativity, clustering and the largest strongly connected component (LSCC) are computed as defined in
Section I. Returns are open-to-close returns computed as differences between log ending and beginning prices during each
interval. Volatility is the range between high and low prices, volume is the total number of contracts traded and duration is the
time (in seconds) elapsed between the start and end of the network interval. Effective Spread is equal to twice the square root
of the first order autocovariance of returns over each interval (x10-3). Signed Volume is equal to the buy volume minus the sell
volume. Trade Imbalance is equal to the number of buy trades less the number of sell trades in each interval. Herfindahl
Trades is the Herfindahl index computed using the number of trades traded by each trader over each interval (x10-3).
Centralization
|Centralization|
Assortativity
Clustering
LSCC
-0.736
0.736
0.097
0.003
0.013
33
Table I: Network Variables: Summary Statistics
The sample contains summary statistics for 12,032 networks each comprised of 600
sequential trades in the nearby e-mini S&P 500 futures contract on the CME
Globex platform during August 2009. Centralization, |centralization|,
assortativity, clustering and the largest strongly connected component (LSCC) are
network variables computed as defined in Section I. ADF probability refers to the pvalue of the ADF test for the null of unit root. Terms in [brackets] below
autocorrelation coefficients refer to p-values of the Portmanteau Q-test for no serial
correlation at 1, 5, and 10 lags.
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
ADF probability
Autocorrelation
Lag 1
Autocorrelation
Lag 5
Autocorrelation
Lag 10
Centralization
0.0022
0.0037
0.6499
-0.7356
0.1711
-0.0200
2.9827
0.0000
0.007
[0.462]
0.079
[0.000]
0.060
[0.000]
|Centralization|
0.1373
0.1177
0.7356
0.0000
0.1021
0.9944
4.0422
0.0001
0.003
[0.779]
0.013
[0.000]
0.029
[0.000]
Assortativity
0.0424
0.0321
0.3385
-0.0557
0.0391
1.5562
6.2788
0.0000
0.120
[0.000]
0.074
[0.000]
0.058
[0.000]
Clustering
0.0524
0.0483
0.2250
0.0004
0.0282
0.9605
4.4588
0.0000
0.197
[0.000]
0.085
[0.000]
0.055
[0.000]
LSCC
0.0912
0.0857
0.5217
0.0024
0.0641
0.7500
3.6643
0.0000
0.273
[0.000]
0.132
[0.000]
0.078
[0.000]
34
Table II: Financial Variables: Summary Statistics
The sample contains summary statistics for 12,032 networks each comprised of 600 sequential trades in the nearby e-mini
S&P 500 futures contract on the CME Globex platform during August 2009. Returns are open-to-close network returns
computed as differences between log ending and beginning prices during each interval. Volatility is the range between high
and low prices, volume is the total number of contracts traded and duration is the time (in seconds) elapsed between the start
and end of the network interval. Effective Spread is equal to twice the square root of the first order autocovariance of returns,
Stoll (1978), over each interval (x10-3). Signed Volume is equal to the buy volume minus the sell volume. When the price is
constant (zero returns), buy/sell volume is equal to that of the previous interval. Trade Imbalance is equal to the number of buy
trades less the number of sell trades in each interval. Herfindahl Trades is the Herfindahl index computed using the number of
trades traded by each trader over each interval (x10-3). ADF probability refers to the p-value of the ADF test for the null of unit
root. Terms in [brackets] below autocorrelation coefficients refer to p-values of the Portmanteau Q-test for no serial correlation
at 1, 5, and 10 lags.
Effective
Signed
Trade
Herfindahl
Returns
Volatility
Volume
Duration
Spread
Volume
Imbalance
Trades
13.804
0.0145
Mean
0.0002
0.0621
2628.7
40.775
7.9369
0.0842
18.000
0.0000
Median
0.0000
0.0507
2521.0
28.000
7.9046
0.0836
5171.0
13.000
Maximum
0.1992
0.2901
7706.0
344.00
21.256
0.1306
-4592.0
-8.0000
Minimum
-0.1493
0.0241
1095.0
0.0000
0.0011
0.0242
1219.3
1.5679
Std. Dev
0.0383
0.0191
636.06
37.926
2.1513
0.0126
0.0462
0.0615
Skewness
-0.0116
0.9046
1.9566
1.9630
0.1778
0.1456
2.9539
3.5352
Kurtosis
2.7977
7.0270
11.016
8.1385
3.7118
3.1088
0.0001
0.0001
0.0000
ADF probability
0.0001
0.0000
0.0000
0.0000
0.0000
Autocorrelations:
-0.0121
0.1747
0.5211
0.5483
0.1867
0.1985
-0.0113
0.3796
Lag 1
[0.185]
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.213]
[0.000]
Lag 5
-0.0136
0.1392
0.3378
0.3805
0.1263
0.0364
-0.0080
0.1840
[0.363]
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.073]
[0.000]
Lag 10
0.0027
0.1076
0.2351
0.3299
0.0992
0.0064
0.0054
0.1249
[0.289]
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.115]
[0.000]
35
Table III: Correlations between Financial and Network Variables
This table contains pairwise correlations calculated from 12,032 networks of 600
sequential trades in the nearby e-mini S&P 500 futures contract on the CME
Globex platform during August 2009. Centralization, |centralization|,
assortativity, clustering and the largest strongly connected component (LSCC) are
network variables computed as defined in Section I. Returns are open-to-close
network returns computed as differences between log ending and beginning prices
during each interval. Volatility is the range between high and low prices, volume is
the total number of contracts traded and duration is the time (in seconds) elapsed
between the start and end of the network interval. Effective Spread is equal to twice
the square root of the first order autocovariance of returns, Stoll (1978), over each
interval (x10-3). Signed Volume is equal to the buy volume minus the sell volume.
Trade Imbalance is equal to the number of buy trades less the number of sell trades
in each interval. Herfindahl Trades is the Herfindahl index computed using the
number of trades traded by each trader over each interval. * indicates statistical
significance at the 5 percent level.
Central |Central- Assorta- Cluster-ization
ization|
tivity
ing
LSCC
Panel A:
|Centralization|
0.008
Assortativity
0.001
0.085*
Clustering
-0.001
-0.235*
-0.038*
LSCC
-0.002
-0.142*
Panel B:
-0.550*
0.541*
Returns
0.675*
-0.011
0.013
0.011
0.006
Volatility
-0.025*
0.170*
-0.072*
-0.034*
0.024*
Volume
0.018
0.093*
-0.052*
0.101*
0.153*
Duration
-0.006
-0.193*
-0.011
0.003
-0.015
Effective Spread
-0.003
-0.157*
0.033
-0.250*
-0.318
Signed Volume
0.637*
0.007
-0.004
-0.009
-0.004
Trade Imbalance
0.669*
-0.013
0.015
0.013
0.002
Herfindahl Trades
-0.004
-0.027*
0.067*
-0.621*
-0.559
36
Table IV: Network Variables: p-values for the Null Hypothesis of Granger Non-causality
The sample contains p-values for the null hypothesis of Granger non-causality for vector autoregressions (VARs)
using the time series of 12,032 network variables each comprised of 600 sequential trades in the nearby e-mini S&P
500 futures contract on the CME Globex platform during August 2009. Centralization, |centralization|,
assortativity, clustering and the largest strongly connected component (LSCC) are network variables computed as
defined in Section I. The VARs are estimated using generalized method of moments (GMM) with HAC robust
standard errors and with optimal lag-length of 12 selected using Akaike Information Criterion. The upper right
quadrant displays p-values for pairwise tests of the first column variable Granger-causing the first row variable,
with Total representing the p-value for the first column variable jointly Granger-causing all variables in the first
row. The lower left quadrant displays p-values for pairwise tests of the first row variable Granger-causing the first
column variable, with All representing the p-value for the first row variable jointly Granger-causing all variables in
the first column.
Centralization |Centralization| Assortativity Clustering
LSCC
Total
Centralization
0.59
0.75
0.99
0.65
0.65
|Centralization|
0.18
0.11
0.01*
0.04*
0.00*
Assortativity
0.69
0.09
0.03*
0.00*
0.00*
Clustering
0.54
0.06
0.00*
0.00*
0.00*
LSCC
0.98
0.15
0.00*
0.06
0.00*
All
0.37
0.05*
0.00*
0.01*
0.00*
37
Table V: Financial and Network Variables: p-values for the Null
Hypothesis of Granger Non-causality
The sample contains p-values for the null hypothesis of Granger non-causality for
VARs using the time series of 12,032 network variables each comprised of 600
sequential trades in the nearby e-mini S&P 500 futures contract on the CME
Globex platform during August 2009. Returns, Volatility, Duration, Volume,
Effective Spreads, Signed Volume, Trade Imbalance, and Herfindahl Trades are
computed over each interval. Centralization, Assortativity, Clustering and the
Largest Strongly Connected Component (LSCC) are network variables computed as
defined in Section I. The VARs are estimated using GMM with HAC robust
standard errors and with optimal lag-length of 12 selected using AIC. Each number
represents the p-value for pairwise tests between variables in each column, with
arrows indicating significance at the 5% level. The p-values in parentheses denote
that these four Network Variables jointly Granger-cause Financial Variables with
significance at the 5 percent level labeled *. The p-values in brackets denote that
Financial Variables jointly Granger-cause the four Network Variables with
significance at the 5 percent level labeled †.
Panel A:
Returns
(0.01)*
[0.00]†
0.00
0.00
Centralization
0.00
0.00
0.25
0.05
Assortativity
0.00
0.04
0.88
0.40
0.00
0.00
0.16
0.68
0.05
0.00
Clustering
LSCC
Volatility
(0.00)*
[0.00]†
Panel B:
Duration
(0.00)*
[0.20]
0.00
0.11
Centralization
0.48
0.00
0.01
0.34
Assortativity
0.05
0.03
0.00
0.51
Clustering
0.15
0.00
0.00
0.01
LSCC
0.14
0.00
Volume
(0.00)*
[0.16]
38
Panel C:
Signed Volume
(0.00)*
[0.35]
0.08
0.10
0.90
0.98
0.75
0.68
0.92
0.80
Centralization
Assortativity
Clustering
LSCC
0.84
0.20
0.19
0.01
0.58
0.00
0.37
0.00
0.20
0.02
0.14
0.21
Herfindahl
Trades
(0.00)*
[0.22]
Panel D:
Effective
Spread
(0.00)*
[0.21]
0.00
0.07
Centralization
0.00
0.14
0.25
0.38
Clustering
0.11
0.66
0.00
0.42
LSCC
0.44
0.17
Assortativity
Trade
Imbalance
(0.03)*
[0.18]
39
Table VI: Financial and Liquidity Measures: p-values for the Null
Hypothesis of Granger Non-causality
The sample contains p-values for the null hypothesis of Granger non-causality for
bi-variate VARs estimated with optimal lag-length 6 selected using the AIC over
12,032 sequential networks each comprised of 600 sequential trades in the nearby
e-mini S&P 500 futures contract on the CME Globex platform during August 2009.
Each number represents the p-value for pairwise tests between variables in each
column, with arrows indicating significance at the 5% level. Returns are open-toclose network returns computed as differences between log ending and beginning
prices during each interval. Volatility is the range between high and low prices,
Volume is the total number of contracts traded and Duration is the time (in seconds)
elapsed between the start and end of the network interval. Effective Spread is equal
to twice the square root of the first order autocovariance of returns, Stoll (1978),
over each interval (x10-3). Signed Volume is equal to the buy volume minus the sell
volume. Trade Imbalance is equal to the number of buy trades less the number of
sell trades in each interval. Herfindahl Trades is the Herfindahl index computed
using the number of trades traded by each trader over each interval.
Panel A:
Signed
Volume
0.18
0.10
Returns
0.55
0.81
0.47
0.91
Volatility
0.00
0.71
0.48
0.02
Volume
0.00
0.01
0.02
0.37
Duration
0.00
0.00
Herfindahl
Trades
Panel B:
Effective
Spread
0.72
0.52
Returns
0.15
0.24
0.05
0.00
Volatility
0.60
0.93
0.00
0.00
Volume
0.29
0.10
0.00
0.00
Duration
0.13
0.07
Trade
Imbalance
40
Table VII: Simulation Results - Correlations between Financial and
Network Variables
This table contains pairwise correlations between network variables each calculated
from 12,000 simulated networks of 600 sequential trades as described in Section IV.
Centralization, |centralization|, assortativity, clustering and the largest strongly
connected component (LSCC) are network variables computed as defined in Section
I. Returns are open-to-close network returns computed as differences between log
ending and beginning prices during each interval. Volatility is the range between
high and low prices, volume is the total number of contracts traded and duration is
the time (in seconds) elapsed between the start and end of the network interval.
Effective Spread is equal to twice the square root of the first order autocovariance of
returns, Stoll (1978), over each interval (x10-3). Signed Volume is equal to the buy
volume minus the sell volume. Imbalance is equal to the number of buy trades less
the number of sell trades in each interval. The Herfindahl Trades measure is the
Herfindahl index computed using the number of trades traded by each trader over
each interval. * indicates statistical significance at the 5 percent level.
Central
-ization
|Centralization|
Assortativity
Clustering
LSCC
Returns
0.605*
-0.024*
0.003
-0.000
0.003
Volatility
0.008
0.219*
0.013
0.004
-0.059*
Volume
-0.004
0.067*
0.119*
0.042*
-0.028*
Duration
0.016
0.003
0.012
-0.002
0.003
Effective Spread
0.052*
0.039*
0.031*
0.007
-0.010
Signed Volume
0.299*
0.009
0.043*
0.021*
-0.016
Imbalance
0.722*
0.052*
-0.011
0.000
-0.022*
Herf. Trades
-0.001
-0.138*
0.001
-0.050*
-0.818*
41